We will develop and apply novel geometric algorithms to brain MRIs obtained from the well-characterized Arizona APOE cohort of presymptomatic individuals and the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset. Our preliminary results have shown statistically significant differences in several brain regions between APOE e4 carriers and non-carriers who were either healthy controls or patients with mild cognitive impairment (MCI) in the ADNI. In the current project we will continue developing novel MRI analysis methods using diffusion geometry and hyperbolic conformal geometry. Specifically, we will develop brain grey matter morphology signatures to measure grey matter morphometry changes and will also extend our surface fluid registration method to hyperbolic Poincar disk model to match lateral ventricular surfaces. The proposed system will focus on two brain cortical and subcortical structures: grey matter and lateral ventricle and compute their complete geometric features to measure all AD related atrophy. This may provide an effective way to pinpoint subregional areas of differential vulnerability associated with known neurodegenerative risk and protective factors for late-onset AD. To investigate the reliability and practicality of our method, we seek to 1) validate developed algorithms with established models in an independent cohort of Arizona APOE carriers, 2) explore APOE genetic influence in MCI patients and healthy subjects in the ADNI dataset and younger presymptomatic individuals of Arizona APOE cohort, and 3) explore additional regions of anatomical interest so as to develop MRI biomarkers for the identification of preclinical stage Alzheimer's disease that in turn will facilitate the therapeutic goals of prevention and earlier intervention to delay dementia onset and retard dementia progression.